Ideal quotient

Not to be confused with the quotient of a ring by an ideal.

In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set

$(I:J)=\{r\in R\mid rJ\subseteq I\}$

Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because $KJ\subseteq I$ if and only if $K\subseteq (I:J)$. The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry (see below).

(I : J) is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there is a related notion of the inverse of a fractional ideal.

Properties

The ideal quotient satisfies the following properties:

• $(I:J)=\mathrm {Ann} _{R}((J+I)/I)$ as $R$-modules, where $\mathrm {Ann} _{R}(M)$ denotes the annihilator of $M$ as an $R$-module.
• $J\subseteq I\Leftrightarrow (I:J)=R$ (in particular, $(I:I)=(R:I)=(I:0)=R$)
• $(I:R)=I$
• $(I:(JK))=((I:J):K)$
• $(I:(J+K))=(I:J)\cap (I:K)$
• $((I\cap J):K)=(I:K)\cap (J:K)$
• $(I:(r))={\frac {1}{r}}(I\cap (r))$ (as long as R is an integral domain)

Calculating the quotient

The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if I = (f₁, f₂, f₃) and J = (g₁, g₂) are ideals in k[x₁, ..., x_n], then

$I:J=(I:(g_{1}))\cap (I:(g_{2}))=\left({\frac {1}{g_{1}}}(I\cap (g_{1}))\right)\cap \left({\frac {1}{g_{2}}}(I\cap (g_{2}))\right)$

Then elimination theory can be used to calculate the intersection of I with (g₁) and (g₂):

$I\cap (g_{1})=tI+(1-t)(g_{1})\cap k[x_{1},\dots ,x_{n}],\quad I\cap (g_{2})=tI+(1-t)(g_{2})\cap k[x_{1},\dots ,x_{n}]$

Calculate a Gröbner basis for $tI+(1-t)(g_{1})$ with respect to lexicographic order. Then the basis functions which have no t in them generate $I\cap (g_{1})$.

Geometric interpretation

The ideal quotient corresponds to set difference in algebraic geometry. More precisely,

• If W is an affine variety (not necessarily irreducible) and V is a subset of the affine space (not necessarily a variety), then

$I(V):I(W)=I(V\setminus W)$

where $I(\bullet )$ denotes the taking of the ideal associated to a subset.

• If I and J are ideals in k[x₁, ..., x_n], with k an algebraically closed field and I radical then

$Z(I:J)=\mathrm {cl} (Z(I)\setminus Z(J))$

where $\mathrm {cl} (\bullet )$ denotes the Zariski closure, and $Z(\bullet )$ denotes the taking of the variety defined by an ideal. If I is not radical, then the same property holds if we saturate the ideal J:

$Z(I:J^{\infty })=\mathrm {cl} (Z(I)\setminus Z(J))$

where $(I:J^{\infty })=\cup _{n\geq 1}(I:J^{n})$.

Examples

• In $\mathbb {Z}$, $((6):(2))=(3)$
• In algebraic number theory, the ideal quotient is useful while studying fractional ideals. This is because the inverse of any invertible fractional ideal $I$ of an integral domain $R$ is given by the ideal quotient $((1):I)=I^{-1}$.
• One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let $I=(xyz),J=(xy)$ in $\mathbb {C} [x,y,z]$ be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in $\mathbb {A} _{\mathbb {C} }^{3}$. Then, the ideal quotient $(I:J)=(z)$ is the ideal of the z-plane in $\mathbb {A} _{\mathbb {C} }^{3}$. This shows how the ideal quotient can be used to "delete" irreducible subschemes.
• A useful scheme theoretic example is taking the ideal quotient of a reducible ideal. For example, the ideal quotient $((x^{4}y^{3}):(x^{2}y^{2}))=(x^{2}y)$, showing that the ideal quotient of a subscheme of some non-reduced scheme, where both have the same reduced subscheme, kills off some of the non-reduced structure.
• We can use the previous example to find the saturation of an ideal corresponding to a projective scheme. Given a homogeneous ideal $I\subset R[x_{0},\ldots ,x_{n}]$ the saturation of $I$ is defined as the ideal quotient $(I:{\mathfrak {m}}^{\infty })=\cup _{i\geq 1}(I:{\mathfrak {m}}^{i})$ where ${\mathfrak {m}}=(x_{0},\ldots ,x_{n})\subset R[x_{0},\ldots ,x_{n}]$. It is a theorem that the set of saturated ideals of $R[x_{0},\ldots ,x_{n}]$ contained in ${\mathfrak {m}}$ is in bijection with the set of projective subschemes in $\mathbb {P} _{R}^{n}$. This shows us that $(x^{4}+y^{4}+z^{4}){\mathfrak {m}}^{k}$ defines the same projective curve as $(x^{4}+y^{4}+z^{4})$ in $\mathbb {P} _{\mathbb {C} }^{2}$.

References

• Viviana Ene, Jürgen Herzog: 'Gröbner Bases in Commutative Algebra', AMS Graduate Studies in Mathematics, Vol 130 (AMS 2012)

• M.F.Atiyah, I.G.MacDonald: 'Introduction to Commutative Algebra', Addison-Wesley 1969.